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A city ordinance requires that a smoke detector be installed in all residential housing. There is concern that too many residences are still without detectors, so a costly inspection program is being contemplated. Let \(p\) be the proportion of all residences that have a detector. A random sample of 25 residences is selected. If the sample strongly suggests that \(p<.80\) (less than \(80 \%\) have detectors), as opposed to \(p \geq .80\), the program will be implemented. Let \(x\) be the number of residences among the 25 that have a detector, and consider the following decision rule: Reject the claim that \(p=.8\) and implement the program if \(x \leq 15\). a. What is the probability that the program is implemented when \(p=.80 ?\) b. What is the probability that the program is not implemented if \(p=.70\) ? if \(p=.60 ?\) c. How do the "error probabilities" of Parts (a) and (b) change if the value 15 in the decision rule is changed to 14

Step by step solution

01

Compute the probability for part a

The claim is that \(p=0.80\). The decision to reject this claim and implement the program is made if \(x \leq 15\). Hence we need to compute the probability \(P(X \leq 15)\) where X follows a binomial distribution with parameters n=25 (size of the sample) and \(p=0.8\). This can be calculated using cumulative binomial probability.

02

Compute the probability for part b

Here, two probabilities have to be calculated: firstly when \(p=0.70\) and secondly when \(p=0.60\). Now the criteria is that the program is not implemented if \(x > 15\). So the probabilities in this case will be \(P(X > 15)\) where X follows binomial distribution. First we shall calculate it for \(p=0.70\) and then \(p=0.60\).

03

Calculate the changes in probabilities for part c

For part c, we will have to compute the error probabilities as in steps 1 and 2, but now the decision threshold is changed from 15 to 14. So the probabilities that need to be calculated are \(P(X\leq14)\) for \(p=0.80\) and \(P(X>14)\) for \(p=0.70\) and \(p=0.60\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Probability Theory

Probability theory is the mathematical framework that allows us to analyze random events and quantify the likelihood of these events occurring. At its core, it deals with understanding and predicting the outcomes of uncertain processes. In our exercise, we look at the probability of a random sample of residences having smoke detectors installed.

Given a population where a certain percentage, denoted by the variable 'p', has smoke detectors, we used a random sample to make inferences about the entire population. This scenario perfectly illustrates the application of probability theory, as it relies on the concept of a binomial distribution—an essential part of probability theory which describes the number of successes in a sequence of independent experiments.

Cumulative Probability

Cumulative probability refers to the probability that a random variable is less than or equal to a certain value. It is essentially the sum of the probabilities of the outcomes up to a specific point and it plays a principal role in evaluating scenarios where 'less than or equal to' conditions are considered.

In the context of our exercise, the cumulative probability is used to calculate the likelihood that 15 or fewer out of 25 residences have a smoke detector installed (when 'p' is hypothesized to be 0.80). This is determined by summing up the probabilities of having 0, 1, 2, ..., up to 15 detectors in our binomial distribution scenario.

Decision Rule

A decision rule in statistics is a pre-determined plan that dictates under which conditions a hypothesis will be rejected or not. It provides clear-cut instructions on how to proceed based on the data. In our smoke detector example, the decision rule states that the claim that 80% of residences have smoke detectors should be rejected if 15 or fewer residences in the random sample have them installed.

Thus, a decision rule serves as the bridge between statistical analysis and actionable outcomes. It's what enables decision-makers to implement policies, like the costly inspection program in our example, based on statistical evidence rather than guesswork.

Statistical Hypothesis Testing

Statistical hypothesis testing is a method used to determine if there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. This process starts by stating two hypotheses: the null hypothesis (denoted as H0) and the alternative hypothesis (H1 or Ha).

In our scenario, the null hypothesis would be that 80% or more residences have smoke detectors (H0: p ≥ 0.80), and the alternative hypothesis would be that fewer than 80% have them (Ha: p < 0.80). The exercise deals with testing these hypotheses by observing a sample and making decisions based on the calculated probabilities. Essentially, hypothesis testing allows us to make probabilistic conclusions about population parameters based on sample statistics.